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The twisted group algebra of a finite nilpotent group over a number field

Published online by Cambridge University Press:  18 May 2009

Hans Opolka
Hans Opolka
Affiliation:
Mathematisches Institut der Universität Münster, Roxelerstrasse 64, 4400 Münster
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Let G be a finite group with neutral element e which operates trivially on the multiplicative group R* of a commutative ring with identity 1. Let H2(G, R*) = Z2(G, R*)/B2(G, R*) denote the second cohomology group of G with respect to the trivial G-module R*. With every factor system (2-cocycle) f ∈ Z2(G, R*) we associate the so called (central) twisted group algebra (R, G, f) of G over R (see [4, Chapter V, 23.7] or [13, §4] for a definition). If f is cohomologous to f', then the R-algebras (R, G, f) and (R, G, f′) are isomorphic. Hence, up to R-algebra isomorphism, (R, G, f) is determined by the cohomology class f∈H2(G, R*) determined by f. If R = k is a field of characteristic not dividing the order |G| of G, then a computation of the discriminant of (k, G, f) shows that (k, G, f) is semisimple (see [13, 4.2]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 20 , Issue 1 , January 1979 , pp. 55 - 61
Copyright
Copyright © Glasgow Mathematical Journal Trust 1979

References

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